Exhibit 99.436
DYNAMIC GAME AND BIDDING STRATEGIES IN DEREGULATED ELECTRICITY
Ali Keyhani
EXECUTIVE SUMMARY
As the demand for electric energy grows, more power producers and power
marketers will enter the energy market. The need for efficient market monitoring
to deal with anti-competitive behavior should be addressed. The solution of this
problem will assure the efficient production of electric energy and stable
operation of the power network. With implementation of emerging metering
technology, the spot price of energy can be sending to the customer. The
metering system can control various loads at the customer sites. The customer
can be offered a number of variable price schedules based on the time of usage.
With the customer in the loop of energy usage and reacting to the spot price
market, control of the ancillary services will need to be investigated. The
first type of result expected in this research is a new understanding of the
behavior of future complex electric energy networks when the interacting
strategies of multiple and players are considered explicitly. System and market
variables observed and monitored by a participant such as an energy provider or
an independent system operator would generally change when the participant
changes its own input to the system. One component of the change is attributable
to the direct effect of the participant's change of input, and this is naturally
expected. A second component of the system change is due to the changes in
inputs of the other participants, as a reaction to the change in input of the
participant. When one does not explicitly consider the presence of the other
participants, one may be led to an erroneous conclusion about the estimated
change in system variables, as caused by a change in a participant's input. The
application of game theory to the study of complex electric energy networks
automatically focuses on the anticipated reactions by other participants. The
second type of expected result is a new insight on the effectiveness of ISO as a
market maker to induce the multiple, independent market players in complex
electric energy systems, to promote greater system-wide benefits as the players
continue to pursue their separate goals. Various market signals will be explored
for incorporation into the incentive mechanisms. The ISO as a market maker can
provide incentive strategies that expected to benefit both the bulk power market
and the ancillary services market. Finally, A MATLAB test bed for concept
evaluation will be developed that can be used as research tool for study of
market dynamics.
2. PROBLEM STATEMENT
Many aspects of price dynamics and bidding strategies in deregulated power
systems have been studied. However, many problems remain to be investigated
where the dynamic power system can be modeled as a dynamic game. In this
research proposal the following problems will be studied:
1. Developing optimal bidding strategies for electricity markets
participants,
2. Different types of gaming in deregulated electricity markets and their
effects on market efficiency and stability,
3. Developing a mathematical framework for the ISO as a market maker to avoid
market gaming and to maximize the power system performance,
4. The effect of load management in pricing of ancillary reserves,
5. The effect of combining transmission congestion contract (TCC) markets
with the ancillary services markets on congestion management improvement.
The above problems will be studied by designing a market simulator and using the
California power system as a case study.
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4. PROBLEM DESCRIPTION AND FORMULATION
Mathematical algorithms such as discrete-time optimal control theory and
feedback Stackelberg game theory (stochastic leader-follower game theory) could
be used to develop optimal bidding strategies for both power suppliers and
consumers in electricity markets. Using discrete-time optimal control theory and
estimating future market clearing prices, each market participant (such as a
Genco or Esco) could control the slope of its linear bid function in order to
maximize its expected payoff in a competitive electricity market for the trading
period T (1 hour in hour-ahead and 24 hours in day-ahead markets). Fuzzy logic
is a powerful mathematical tool in developing and implementing game-theoretic
bidding strategies in competitive electricity markets. By using fuzzy logic
method a market participant could study the effect of random variables such as
fuel price, system residual demand, transmission system reliability factor and
market clearing price on optimizing its bidding strategy and maximizing its
expected payoff in the market. By trying different types of fuzzy membership
functions and using Takagi-Sugeno fuzzy systems, we can develop variety of
bidding strategies and study their efficiency and applicability in electricity
markets. The development of a market simulator for comparing the effect of
different bidding strategies on electricity markets stability and efficiency
seems necessary. One could study the effect of power system constraints such as
contingencies, transmission capacity limits (transmission congestion), voltage
stability limits and transient stability limits on strategic bidding by
integrating the optimal power flow algorithm to the virtual market simulator.
This could be made possible by using powerful CAD tools, such as Matlab, for
computational aspects of the market simulator and then interface the
computational programs with graphical user interface (GUI) programs, such as
Visual Java, to show the market simulation results to the market players.
Multi-market systems, such as deregulated California power system, have the
advantage of allowing generators and loads to choose their preferred auction
mechanism. The decision to allow competing markets has also some drawbacks:
- - The existence of multiple markets reduces the supply elasticity (price
responsive) of each market, making each market more susceptible to gaming.
- - Power suppliers can unexpectedly shift portions of their generation
between markets. This allows them to drive up prices in selected markets
without having to pay the opportunity cost of letting the withheld
capacity stand idle.
When considering this type of inter-market gaming, it is important to keep in
mind the time frame during which the market power can be exerted. If one power
exchange consistently recorded a higher market-clearing price, generators from
competing exchanges could simply increase their profits by shifting their bids
into the high priced market until the prices equalize. Consider the PX day-ahead
market. In this market bids are supplied to the PX for all hours of the day
simultaneously. A generator intending to game this market may choose a set of
hours for the day to for which he shifts a portion of his bid to a different
power auction. Since the PX auction is carried out for all hours of the day
simultaneously, there is no chance for competing generators to react to this
shift until the following day, during which the gaming generator may have
altered his strategy again. As we can see in this example, the efficiency of the
market is heavily reliant on the speed by which economic feedback is provided to
participants. The opening of the hour-ahead market on the California power
exchange was an important step in improving the market response time. Under the
current system of power trading in California, suppliers can choose between a
numbers of time frames in which to bid their generation for any given hour of
operation. In addition to bilateral markets (such as BFM in PX), for which
contracts can be specified months in advance, there exists the PX day-ahead, PX
hour-ahead and ISO real-time auctions. Just as with competing power exchanges,
the time separated auctions have to be in a long-term equilibrium. For example,
higher prices in the day-ahead compare to the hour-ahead market
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would cause generators and loads to shift their bidding patterns until the price
discrepancy is equalized. As was the case with inter-market gaming, the
self-adjusting process of the market is relatively slow to respond to
inter-temporal markets. This opens the door for strategic bidders to exploit
short-term price imbalances between temporal markets. The process of
inter-temporal gaming is quite similar to the inter-market gaming. A supplier
can temporarily drive up prices in one market by suddenly shifting a portion of
its bids from one temporal auction to another. An interesting fact about
inter-temporal games is that the bidder can act as a supplier and consumer for
the same operating hour. A supply bid offered in the day-ahead market can be
partially or fully repurchased in the hour-ahead market. The existing auction
mechanism does not distinguish between a load bid and a generation repurchase.
This potentially allows generators to misrepresent their intentions in the
day-ahead market, by either over or underbidding their resources, and adjusting
the bids in the hour-ahead and real-time markets. Similarly load aggregators can
choose to underbid their demand functions on the day-ahead market in order to
drive down the market-clearing price.
In this research, feedback Stackelberg game theory will be used to study the
behavior of market participants in deregulated electricity markets and develop a
mathematical framework for the ISO (as a market maker) to interact properly with
other market participants' strategies in order to avoid market gaming and
maximizing the power system performance. In order to assure system security and
comply with FERC standards, the California ISO runs markets for generation
reserve. Reserve requirements are separated into four categories: (1) Spinning
reserve, (2) Non-spinning reserve, (3) Replacement reserve, and (4) Regulation
reserve. As a whole they ensure that the system operator always has sufficient
generation resources to dispatch in the case of outages or other contingencies
as well as imbalances caused by load forecast errors. The reserve contracts are
compensated using a two-part pricing system. First, generators are paid for
keeping part of their capacity for dispatch. Second, if this capacity is in fact
used, then the generators receive a second payment corresponding to their energy
bids. The division of reserves into four distinct categories was motivated by a
need to accurately pass on the cost of reserves to market participants. Each
category corresponds to a different type of contingency, and depending on the
type of transaction the scheduling coordinator (such as PX) files with the ISO,
it will be assigned a specific portion of the reserve requirements from each
reserve market. The result of this division was creating a number of relatively
small sub-markets, each sizing from 1,000 MW to 3,000 MW for every trading hour.
The small size of the ancillary services markets in California system make them
more vulnerable to gaming by large power suppliers. In this research work, the
effect of combining these four separated ancillary services markets into one
reserve market on improving the market efficiency and stability will be studied.
The effect of binding the transmission congestion contract (TCC) market into the
ancillary services market on congestion management improvement will also be
studied with the aid of a market simulator. Discrete-time optimal control theory
could be used by power suppliers (Gencos) and consumers (Escos) in order to
optimize their bidding strategies in double auctions. A discrete time
optimization problem could be formulated for a power supplier as following:
We assume that each power supplier has a linear bid function (LBF) and a
quadratic cost function with respect to its output power:
[FORMULA OMITTED] , Generator's quadratic cost function (4.1)
[FORMULA OMITTED] , Generator's linear bid function (LBF) (4.2)
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[FORMULA OMITTED] , Generator's marginal cost function (MCF) (4.3)
As it was discussed in [3,11,12], each generator will try to maximize its
revenue by increasing its output power in order to win the auction at the
market-clearing price ((lambda)j). Each Genco could match up its marginal cost
with the market-clearing price by selecting optimal bidding strategies during
the trading period in order to maximize its profits. Its optimal bidding
strategy could be increasing its generation level or increasing the slope of its
linear bid function (LBF) or both. A candidate profit maximization problem for a
Genco could be formulated as follows:
Maximize [FORMULA OMITTED] (4.4)
By using equations (4.2) and (4.3) the above maximization problem could be
rewritten as:
Maximize [FORMULA OMITTED] (4.5)
Subject to:
[FORMULA OMITTED] (4.6)
[FORMULA OMITTED] (4.7)
Where,
J[g] = Total cost of generation for time period T,
(alpha)gj = Generator's variable cost,
(beta)gj = Generator's start-up and no-load costs,
Pg[j] = Generator's output power,
(lambda)j = Market clearing price at time j,
(lambda)gj = Generator's marginal cost,
u[gj] = Generator's bidding strategy.
The solution for the above optimization problem based on proposition 1 (see
Appendix I) is as follows:
[FORMULA OMITTED], Hamiltonian function (4.8)
[FORMULA OMITTED] (4.9)
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[FORMULA OMITTED] (4.10)
Where, (gamma)j is the convergence step size in the gradient method.
[FORMULA OMITTED] (4.11)
[FORMULA OMITTED] (4.12)
If a Genco participates in a day-ahead energy market, then its optimal
bidding strategies and generation schedules for the next 24 period would be:
[FORMULA OMITTED]
If a Genco participates in an hour-ahead energy market, the optimal bidding
strategy would be to bid at the market-clearing price through all the trading
hours. Therefor, the optimal bidding strategies and generation schedules could
be derived from the following:
[FORMULA OMITTED] (4.13)
[FORMULA OMITTED] (4.14)
[FORMULA OMITTED] (4.15)
A similar optimization problem can be formulated for a power consumer who
participates in a day-ahead or Hour-Ahead energy market. We assume that each
power consumer has a linear bid function (LBF) and a quadratic benefit function
with respect to its demand:
[FORMULA OMITTED] , Consumer's quadratic benefit function (4.16)
[FORMULA OMITTED] , Consumer's linear bid function (LBF) (4.17)
[FORMULA OMITTED] , Consumer's marginal benefit function (MBF) (4.18)
Each Esco could match up its marginal benefit with the market-clearing
price by selecting optimal bidding strategies during the trading period in order
to maximize its benefits. Its optimal bidding strategy could be increasing its
demand level or decreasing the slope of its linear bid function (LBF) or both. A
candidate benefit maximization problem for an Esco could be formulated as
follows:
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Maximize [FORMULA OMITTED] (4.19)
Subject to:
[FORMULA OMITTED] (4.20)
[FORMULA OMITTED] (4.21)
Where,
J[d] = Total consumer's benefit for time period T,
(alpha)dj = Consumer's variable benefit,
(beta)dj = Consumer's no-demand benefit,
Pd[j] = Consumer's demand,
(lambda)j = Market clearing price at time j,
(lambda)dj = Consumer's marginal benefit,
u[dj] = Consumer's bidding strategy.
The bidding strategy (udj) does not appear in the objective function
(4.19). Therefore, it seems to be that the optimal bidding strategy for an Esco
that participates in either day-ahead or hour-ahead energy market is to bid at
market clearing price through all the trading hours. The optimal bidding
strategies and demand schedules could be derived as following:
[FORMULA OMITTED] (4.22)
[FORMULA OMITTED] (4.23)
[FORMULA OMITTED] (4.24)
For an Esco that participates in a day-ahead energy market, optimal bidding
strategies and demand schedules for the next 24 period would be:
[FORMULA OMITTED]
The main objective of the ISO is to optimize the price of the electricity
used by the consumers in a deregulated power system with respect to the power
system security and reliability constraints. The ISO collects the supply bids
and capacity schedules and the demand bids and load schedules from the schedule
coordinator (PX in California) together with the market clearing price. The PX
constructs aggregated hourly supply and demand curves to determine market
clearing prices as well as
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corresponding supply and demand schedules. According to the PX bidding protocol,
a marginal clearing price is set at the intersection point between the
aggregated demand and supply curves for each of the 24 scheduling hours. All the
generators winning the auction are paid the market clearing price and all the
loads are billed at the market-clearing price. The schedule adjustment bids for
a power supplier (consumer) represents the desire of the market participant to
increase (decrease) its schedule if energy price increases (decreases) or
decrease (increase) its schedule if the energy price decreases. Participants
that provide SABs indicate that are willing to participate in the congestion
management and pricing process (TCC auction). Participants who do not provide
SABs are price takers as far as congestion charges are concerned. They want
their schedules to remain constant and are willing to pay any congestion charges
declared by the ISO.
The ISO minimization problem in order to find the optimum generation schedules
and congestion prices could be formulated as following:
[FORMULA OMITTED] (4.25)
Subject to:
[FORMULA OMITTED] , for k =1,...,24
[FORMULA OMITTED] , for k =1,...,24
[FORMULA OMITTED] , for k =1,...,24
[FORMULA OMITTED] , for k =1,...,24
Where, P(k) denotes a (N-1) x 1 (N is the no. of busses in the system) nodal
power injection vector (excluding the reference bus) and H denotes the transfer
admittance matrix (dimension M x (N-1) ) that relates the nodal power injections
to branch power flows (The elements of H are the line distribution factors, see
Appendix IV for more details). z(max) is a M x 1 vector of power flow limits
that could vary by time, depending on the weather conditions and system status
(e.g. line or transformer outage), and M is the no. of transmission lines in the
system.
m = No. of generators participating in the PX day-ahead market
n = No. of loads participating in the PX day-ahead market
mcp(k) = Market clearing price at hour k
[FORMULA OMITTED] = The i(th) generator's supply bid at hour k
[FORMULA OMITTED] = The i(th) generator's schedule at hour k
[FORMULA OMITTED] = The i(th) load's demand bid at hour k
[FORMULA OMITTED] = The i(th) load's schedule at hour k
[FORMULA OMITTED] (4.26)
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[FORMULA OMITTED], Transmission system loss factors (4.27)
The above Kuhn-Tucker optimization problem could be solved using Lagrangian
relaxation,
[FORMULA OMITTED] (4.28)
Where [FORMULA OMITTED],[FORMULA OMITTED], [FORMULA OMITTED]and[FORMULA
OMITTED]are the Lagrange-multipliers (co-states), which are non-negative and:
[FORMULA OMITTED], for i =1,...,m and j = 1,..., M (4.29)
[FORMULA OMITTED], for i =1,...,n and j = 1,..., M (4.30)
[FORMULA OMITTED], for i =1,...,m and j = 1,..., M (4.31)
Different mathematical methods (such as Gradient, Newton,...) could be used to
solve the above optimization problem for the ISO as a market maker. Suppose that
the solution of the above minimization problem is given as follows:
[FORMULA OMITTED]
[FORMULA OMITTED]
[FORMULA OMITTED]
The above optimal generation and load schedules satisfy the transmission
system constraints and demand requirements. Therefore, these optimal schedules
will guarantee a power system security and stability. One proposed method for
the ISO as a market maker to avoid inter-temporal market gaming
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is to use incentive functions as a means of economic signals to the hour-ahead
market participants. One of the proposed forms for the above mentioned economic
incentive functions (for each trading hour k in an hour-ahead market) is as
following:
[FORMULA OMITTED] (4.32)
Where,
[FORMULA OMITTED], for i = 1,2,...,m (4.33)
[FORMULA OMITTED], for i = 1,2,...,n (4.34)
The economic incentive functions (EIF) of the form (4.32) will give economic
signals to the market participants that if they include these functions in their
bid/schedule optimization for the hour-ahead markets, the security and optimal
performance of the power system is guaranteed by the ISO. However, if they avoid
using these EIFs in their bidding strategies for the hour-ahead markets, they
will have to pay transmission congestion charges to the ISO for each hour of
trading electricity that will reduce the market efficiency. We need to develop a
mathematical framework for the ISO as a market maker to reduce inter-temporal
market gaming. It is intended to develop different EIFs and study their effects
on improving the electricity markets efficiency and stability in this research
work.
6. VIRTUAL MARKET SIMULATOR
For experimentally testing various power exchange auction markets using human
decision makers, an interactive and web-based market simulator is needed to
simulate competitive electricity markets in the context of a deregulated power
industry. In this research, a virtual market simulator will be designed and
developed in order to study different aspects of market gaming in deregulated
electricity markets. This simulation environment will also be used to develop
and test different bidding strategies for California markets participants. It
will also be used to develop a mathematical framework for the ISO as a market
maker to interact properly with other market participants' strategies in order
to avoid market gaming and maximizing the power system performance. The
structure of a proposed market simulator is shown in figure 5.1.
The virtual market simulator consists of the following modules:
1. OPF Solver Engine: This module will solve the ISO optimal power flow
problem.
2. Incentive Function Calculator: This module will calculate proper incentive
functions for the market participants.
3. Optimal Bid Calculator: This module will calculate optimal bidding
strategies for the market participants.
4. Adaptive Fuzzy Logic Estimator: This module will estimate future market
clearing prices, transmission system reliability indices, system demand
and ancillary reserves.
5. Database: This module will collect the necessary data from the market
participants (such as their bids and schedules) and also historical data
about the power system status (transmission contingencies, market clearing
prices, ...) through the ISO/PX public access databases.
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6. GUI: This module provides a friendly interface for users to interact with
the virtual market simulator. It performs two functions, display of
information and retrieval of user requests from individual participants to
the server computer via Internet. The OPF solver engine, Incentive
function calculator, optimal bid calculator, and Adaptive fuzzy logic
estimator will be developed using Matlab. The Database and GUI modules
will be developed by Visual Java. It is intended to design a market
simulator that can handle calculations for a power system with 30
zones/nodes, 50 transmission lines, 50 power transactions (generators,
loads, energy brokers and load serving entities). The California
deregulated power system will be used as a case study.
[FLOW CHART]
Figure 5.1 A Market Simulator Structure
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6. WORK PLAN
The following tasks will be implemented in this research work:
Task 1: Design and development of a virtual market simulator,
Task 2: Development of optimal bidding strategies for electricity markets
participants,
Task 3: Study different types of gaming in deregulated electricity
markets and their effects on market efficiency and stability. The
following market gaming will be studied:
A) Intra-Market Gaming: Gaming within a single isolated power market. The
opportunity cost of gaming this market is given by the difference of price
and marginal cost as described by the market power index (IMP).
B) Inter-Market Gaming: Gaming between separate power exchanges or scheduling
coordinators. This may allow a generator to reduce the opportunity cost of
withholding generation from the market by shifting that portion of the bid
to a different power exchange or into bilateral contract.
C) Inter-Commodity Gaming: Similar to inter-market gaming except excess
generation capacity is shifted into ancillary services market.
Inter-Temporal Gaming: Load and generation are shifted between day-ahead,
hour-ahead and real-time markets
Task 4: Development of a mathematical framework for the ISO as a market
maker to avoid market gaming and to maximize the power system
performance,
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Task 5: The effect of load management in pricing of ancillary reserves,
Task 6: The effect of combining transmission congestion contract (TCC)
markets with the ancillary services markets on congestion management
improvement.
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